How are Penrose tilings related to fivefold symmetry in quasicrystals?

Penrose tilings serve as a two-dimensional model that explains how quasicrystals can exhibit fivefold rotational symmetry while maintaining long-range order, despite lacking the translational periodicity found in ordinary crystals.12

In standard crystallography, fivefold symmetry is forbidden because it is incompatible with the translational periodicity required for a repeating unit cell. However, quasicrystals resolve this conflict through quasiperiodicity. The Penrose pattern demonstrates this by using specific shapes arranged in a nonrepeating, deterministic array.123

Key relationships include:
* The Golden Mean: The spacing between rows in a Penrose tiling follows a sequence of large (L) and small (S) intervals, where the ratio of their widths is the golden mean ($\tau$).12
* Fibonacci Sequence: The arrangement of these large and small intervals follows the Fibonacci sequence, a hallmark of quasiperiodic order.2
* Diffraction: Just as Penrose tilings produce diffraction patterns with fivefold symmetry, the atomic structures of quasicrystals produce similar patterns, confirming their long-range order despite the absence of a repeating unit cell.12